The inverse estimation method was originally proposed by Wang (1992) and was applied to study parameter estimation for Weibull distribution. Different from the regular method of moments, the idea of the inverse moment estimation (IME) is as follows.

For a sample X₁, …, X_n from a distribution with unknown parameters, first transform the original sample to a quasi-sample Y₁, …, Y_n, where Y_i contains the unknown parameters but its distribution does not depend on unknown parameters, that is, Y_i is a pivot variable, i = 1, …, n. The population moments of the new sample do not dependent on unknown parameters while the sample moments do. Let the population moments of the quasi-sample equal the sample moments and solve the unknown parameters.

Wang (2004) obtained the inverse moment estimators and the interval estimation based on type II progressively censored data under the Weibull distribution. The simulation results showed that the mean square errors of the inverse moment estimators are less than the maximum likelihood estimates (MLE)’s. Gu and Yue (2013) considered the problem of estimating parameters of the generalized exponential distribution based on a complete sample. They proposed the inverse moment estimators of the parameters of the generalized exponential distribution. The precisions of MLEs and IMEs are compared through numerical simulations. Gui (2015) studied the problem of estimating unknown shape and scale parameters of exponentiated half logistic distribution. Inverse moment and modified inverse moment estimators were derived. Monte Carlo simulations were conducted to compare the performances.

Wang et al. (2010) obtained some exponential inequalities for a negatively orthant dependent sequence and used the exponential inequalities to study the asymptotic approximation of an inverse moment for negatively orthant dependent random variables. Ye and Yang (2013) proposed a new method for dimension reduction in regression using the first two inverse moments. Yang et al. (2014) discussed the asymptotic approximation of an inverse moment for nonnegative random variables. Cheng et al. (2014) established inverse moment bounds for sample autocovariance matrices based on a detrended time series.

In this paper, we focus on the problem of parameter estimation for the inverted exponential Pareto distribution and demonstrate the inverse estimation and its modified version. We begin with the classical MLE and obtain a necessary and sufficient condition for the existence and uniqueness of MLE of the parameters. We propose inverse moment and modified inverse moment estimators and study their properties. Monte Carlo simulations are used to compare the performances. We also propose the methods for constructing joint confidence regions for the two parameters and study their performances.

Gupta et al. (1998) introduced the exponential Pareto distribution Y to model failure time data. The probability density function of Y is given by

where α > 0 and λ > 0 are two parameters. The corresponding cumulative distribution function is

A random variable X = 1/Y is said to have the inverted exponential Pareto distribution. Its cumulative distribution function (cdf) and probability density function (pdf) are specified by

and

respectively, where λ > 0 and α > 0 are the parameters. We denote this distribution as IEPD(λ, α). When α = 1, the inverted exponential Pareto distribution reduces to the inverted Pareto distribution. In this paper, we will show that the two-parameter inverted exponential Pareto distribution can be quite effective in modeling lifetime data.

Shawky and Abu-Zinadah (2009) considered the maximum likelihood estimation of the different parameters of an exponential Pareto distribution. Afify (2010) obtained Bayes and classical estimators for two parameters exponentiated Pareto distribution when a sample is available from complete, type I and type II censoring scheme. Ali et al. (2010) derived the distribution of the ratio of two independent exponentiated Pareto random variables and studied its properties. Singh et al. (2013) proposed maximum likelihood estimators and Bayes estimators of parameters of exponentiated Pareto distribution under general entropy loss function and squared error loss function for progressive type-II censored data with binomial removals.

The rest of this paper is organized as follows. In Section 2, we discuss the classical maximum likelihood estimation of the parameters of the inverted exponential Pareto distribution. In Section 3, we propose the inverse and modified inverse estimation methods to estimate the parameters and study their properties. Joint confidence regions for the two parameters are also proposed in Section 4. Section 5 conducts simulations to compare the estimators and the confidence regions. In Section 6, a numerical example is presented to illustrate the superiorities of the proposed methods. Finally, Section 7 concludes.

In this section, we discuss the MLEs of the parameters of inverted exponential Pareto distribution (IEPD) based on a complete sample. Let X₁, X₂, …, X_n be a random sample from IEPD(λ, α) with pdf and cdf as (1.4) and (1.3), respectively. The log-likelihood function is given by

The score equations are as:

Consider the case when x₁ = · · · = x_n = x, the MLEs λ̂ and α̂ are

and λ̂ is the solution of the equation

where

Note that

since (t − 1)² > (log t)²t, for all t > 1.

since (1/t) log(1 − t) + 1 < 0, for all t > 0. J′(λ) < 0, J(λ) is a decreasing function of λ. Moreover, we have J(0) = ∞ and J(∞) = 0. Thus, J(λ) has no roots in the interval (0,∞).

In the following, we discuss the existence and uniqueness of MLEs in the case of at least two non-identical observed values of the sample.

Theorem 1

Let X₁, X₂, …, X_nbe a random sample from IEPD(λ, α), if the observed values of the sample are not identical, that is x_i ≠ x_(n) = max{x₁, …, x_n} for at least one i ∈ {1, …, n − 1}, then MLEs of λ and α exist and unique.

In Statistics, there are many methods available for estimating the parameter(s) of interest. One of the oldest methods is the method of moments. It is based on the assumption that sample moments should provide adequate estimates of the corresponding population moments. Suppose we want to estimate θ = (θ₁, …, θ_k), the procedure is:

• Find k population moments, μ_i, i = 1, 2, …, k. μ_i will include one or more parameters θ₁, …, θ_k.

• Determine the corresponding k sample moments, m_i, i = 1, 2, …, k.

• Let μ_i = m_i, i = 1, 2, …, k, solve for the parameters. The solution is a moment estimator.

The method of moments is simple and easy to compute. However, the estimator may not be unique or not exist. In this section, we propose an inverse moment estimation. The superiority of the new estimator is its existence and uniqueness.

Definition 1

Suppose X ~ F(x, θ), where θ = (θ₁, …, θ_k) is a parameter vector to be estimated. Transform X to a pivotal variable Y = g(X, θ) whose distribution does not depend on θ. The population momentsμi′ (i=1,…,k)of Y will not contain θ. The sample Y_i = g(X_i, θ) (i = 1, …, k) is called quasi-sample since it is a function of sample (X₁, …, X_n) and parameter θ. The moments of the quasi-samplemi′ (i=1,…,k)is also a function of θ. Letμi′=mi′, i = 1, 2, …, k, solve for the parameters. The solution is an inverse moment estimator.

Let X₁, …, X_n form a sample from IEPD(λ, α) with pdf given in (1.4), it is known that F(X_i), 1 − F(X_i), i = 1, …, n follow uniform distribution U(0, 1), and − log[1 − F(X_i)], i = 1, …, n follow standard exponential distribution Exp(1). By the method of inverse moment estimation, we set

that is,

Thus, the IME of α is obtained as a function of λ,

which is identical with the MLE of α.

Lemma 1

Let Z₍₁₎ ≤ Z₍₂₎ ≤ · · · ≤ Z_(n)be the order statistics from the standard exponential distribution. Then, the random variables W₁, W₂, …, W_n, where

with Z₍₀₎ ≡ 0, are independent and follow standard exponential distributions.

Lemma 2

Let W₁, W₂, …, W_nbe iid standard exponential variables, S_i = W₁ + · · · + W_i, U_i = (S_i/S_i+1)ⁱ, i = 1, 2, …, n − 1, U_n = W₁ + · · · + W_n, then

• U₁, U₂, …, U_nare independent;

• U₁, U₂, …, U_n−1follow the uniform distribution U(0, 1);

• 2U_nfollows χ²(2n).

Lemma 3

The following limits hold: (1) lim_λ→0 log[1 − (a + 1)^−λ]/log[1 − (b + 1)^−λ] = 1, for a > 0, b > 0. (2) lim_λ→∞ log[1 − (a + 1)^−λ]/log[1 − (b + 1)^−λ] = 0, for a > b > 0. (3) lim_λ→∞ log[1− (a + 1)^−λ]/log[1 − (b + 1)^−λ] = +∞, for b > a > 0.

Lemma 4

For t > 0, f(t) = (1 + te^t − e^t)/(1 − e^t) is a decreasing function of t.

Theorem 2

Let W_i = (n−i+1)(Z_(i)−Z_(i−1)), i = 1, 2, …, n form a sample from standard exponential distribution, S_i = W₁ + · · · + W_i, then for t > 0, equation∑i=1n-1log(Sn/Si)=thas a unique positive solution.

Let X₁, X₂, …, X_n form a sample from IEPD(λ, α), X₍₁₎ ≤ X₍₂₎ ≤ · · · ≤ X_(n) are the order statistics from this sample. Let Z_(i) = −α log[1−(1/X_(i)+1)^−λ], i = 1, …, n. Thus, Z₍₁₎ ≤ Z₍₂₎ ≤ · · ·≤ Z_(n) are the first n order statistics from the standard exponential distribution. By Lemma 1, W_i = (n−i+1)(Z_(i)−Z_(i−1)), i = 1, 2, …, n form a sample from standard exponential distribution. Let S_i = W₁ + · · · + W_i, U_i = (S_i/S_i+1)ⁱ, i = 1, 2, …, n − 1, U_n = W₁ + · · · + W_n. Hence,